Difference between revisions of "2020 AIME I Problems/Problem 1"
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In <math>\triangle ABC</math> with <math>AB=AC,</math> point <math>D</math> lies strictly between <math>A</math> and <math>C</math> on side <math>\overline{AC},</math> and point <math>E</math> lies strictly between <math>A</math> and <math>B</math> on side <math>\overline{AB}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | In <math>\triangle ABC</math> with <math>AB=AC,</math> point <math>D</math> lies strictly between <math>A</math> and <math>C</math> on side <math>\overline{AC},</math> and point <math>E</math> lies strictly between <math>A</math> and <math>B</math> on side <math>\overline{AB}</math> such that <math>AE=ED=DB=BC.</math> The degree measure of <math>\angle ABC</math> is <math>\tfrac{m}{n},</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</math> | ||
− | == Solution == | + | == Solution 1== |
<asy> | <asy> | ||
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-molocyxu | -molocyxu | ||
+ | ==Solution 2== | ||
+ | Let <math>\angle{BAC}</math> be <math>x</math>. <math>\angle{ADE}=x</math>. | ||
+ | By Exterior Angle Theorem on triangle <math>AED</math>, <math>\angle{BED}=2x</math>. | ||
+ | By Exterior Angle Theorem on triangle <math>ADB</math>, <math>\angle{BDC}=3x</math>. | ||
+ | This tells us <math>\angle{BCA}=\angle{ABC}=3x</math> | ||
+ | Thus <math>x=\frac{180}{7}</math> and we want <math>\angle{ABC}=3x=\frac{540}{7}</math> to get an answer of <math>\boxed{547}</math>. | ||
==See Also== | ==See Also== |
Revision as of 20:20, 14 March 2020
Contents
[hide]Problem
In with point lies strictly between and on side and point lies strictly between and on side such that The degree measure of is where and are relatively prime positive integers. Find
Solution 1
If we set to , we can find all other angles through these two properties: 1. Angles in a triangle sum to . 2. The base angles of an isoceles triangle are congruent.
Now we angle chase. , , , , , . Since as given by the problem, , so . Therefore, , and our desired angle is for an answer of .
-molocyxu
Solution 2
Let be . . By Exterior Angle Theorem on triangle , . By Exterior Angle Theorem on triangle , . This tells us Thus and we want to get an answer of .
See Also
2020 AIME I (Problems • Answer Key • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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